Comparison of two Euler equation sets in a Discontinuous Galerkin solver for atmospheric modelling (BRIDGE v0.9)

Abstract. The implementation of a 'classical' Discontinuous Galerkin (DG) solver for atmospheric flows is presented that is designed for efficient use in numerical weather prediction, climate simulations, and meteorological research both on the whole sphere and for limited area modeling. To this purpose the horizontally explicit, vertically implicit (HEVI) approach is used together with implicit-explicit (IMEX)-Runge-Kutta (RK) time integration schemes and a moderate spatial approximation order (order 4 or 5). Two Euler equation sets using mass, momentum and either density weighted potential temperature θ or total energy E as prognostic variables are compared by several idealised test cases. Details of the formulation of the Euler equations in covariant form using the Ricci tensor calculus, the linearisations needed for HEVI (especially for the total energy set), boundary conditions for an IMEX-RK scheme, and filtering for numerical stabilisation are given. Furthermore, the implementation of distributed memory parallelisation, the tensor product representation for prismatic grid cells, and optimisations for the HEVI formulation, are outlined. These developments lead to the so-called BRIDGE code, which will serve as a code base for a later DG extension of the well established ICON model. From the used idealised test cases,  which are standard benchmarks for dynamical core development for the atmosphere, we conclude that the equation set using total energy E has better well-balancing properties than the set using θ. This result can be confirmed by a normal mode stability analysis. However, in some tests the set using E suffers more from non-linear instabilities that can only partially solved by filtering.